New dimension bounds for αβ sets

Simon Baker

doi:10.1016/j.jnt.2021.04.018

New dimension bounds for αβ sets

Simon Baker

Mathematics

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Abstract

In this paper we obtain new lower bounds for the upper box dimension of αβ sets. As a corollary of our main result, we show that if α is not a Liouville number and β is a Liouville number, then the upper box dimension of any αβ set is 1. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.

Original language	English
Pages (from-to)	59-72
Journal	Journal of Number Theory
Volume	228
Early online date	31 May 2021
DOIs	https://doi.org/10.1016/j.jnt.2021.04.018
Publication status	E-pub ahead of print - 31 May 2021

Keywords

Diophantine approximation
Self-similar sets
αβ sets

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AB - In this paper we obtain new lower bounds for the upper box dimension of αβ sets. As a corollary of our main result, we show that if α is not a Liouville number and β is a Liouville number, then the upper box dimension of any αβ set is 1. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.

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JO - Journal of Number Theory

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ER -

New dimension bounds for αβ sets

Abstract

Keywords

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